Claims:

1. A computer implemented method for determining values of anisotropic
model parameters of a three dimensional TTI earth model, the anisotropic
parameters including P-wave velocity (Vp0) along a tilted symmetry
axis, the Thomsen anisotropy parameters δ, ε (or
η=(ε-.delta.)/(1+2.delta.) representative of variation of
wave velocities as a function of wave propagation angle from the symmetry
axis, the method comprising: a. obtaining an initial TTI earth model that
substantially flattens common-imaging-point gathers and substantially
ties seismic data to well data, the initial migration velocity model
including initial values Vp0, δ, ε (or η) at each
of a plurality of subsurface locations in a geological volume of
interest; b. inputting checkshot data, or VSP data, or both checkshot and
VSP data at well locations into a three dimensional tomographic inversion
to determine updated values of Vp0 near the well locations, the
values of Vp0 being updated by a correction ΔVp0, wherein
Vp0=Vp0+ΔVp0; c. determining an incremental
improvement Δδ to δ(initial) using the relative change
Δδ=(ΔVp0)/Vp0; d. extrapolating the relative
change Δδ from near-well locations to the entire three
dimensional TTI earth model at each of the plurality of subsurface
locations, taking into account geological consistency and regularization,
to determine updated values of δ, wherein
δ=δ+Δδ; e. determining updated values of
Vp0=Vp0 (1-.DELTA.δ) using the extrapolated three
dimensional Δδ at each of the plurality of subsurface
locations and obtaining the three dimensional extended incremental update
ΔVp0=-.DELTA.δ Vp0; f. inputting
near-to-mid-offset/angle residual moveout information in the
common-imaging-point gathers obtained with an improved migration velocity
model that is defined with the updated values of Vp0 and δ
into a TTI tomographic inversion process to further provide updated
values of δ at each of the plurality of subsurface locations; and
g. inputting near-to-far-offset/angle residual moveout information in the
common-imaging-point gathers obtained with the improved model the TTI
tomographic inversion process to provide updated values of η at each
of the plurality of subsurface locations.

2. The method of claim 1, further comprising iteratively repeating (a),
(b), (c), (d), (e), (f) and (g), wherein the improved parameters of the
TTI model determined at the end of an iteration corresponds to the
initial parameters of the TTI model at the next iteration.

[0002] The present invention relates to a method for determining values of
anisotropic model parameters of a Tilted Transversely Isotropic (TTI)
earth model.

BACKGROUND

[0003] In order to analyze a geological structure of a subterranean
formation, exploration geophysicists make many assumptions. One of them
is that the subterranean formation is isotropic while in fact it is
fundamentally anisotropic. This faulty assumption may result in erroneous
imaging and interpretation of the geological structure. To extend the
seismic processing techniques to anisotropic media, it is desirable to
obtain a measure of the anisotropy of the geological structure.

[0004] Seismic anisotropy can be defined as the dependence of seismic
velocity on the direction of wave propagation. It is known that a
transverse isotropy with tilted axis earth model or TTI earth model can
be used to model the propagation of waves and obtain an image of the
subterranean formation in anisotropic media. The physical parameters to
describe a TTI earth model include (1) the symmetry axis, (2) P-wave
(compressional) velocity along symmetry axis--Vp0, (3) a parameter
that specifies how the velocities vary for small angles from the symmetry
axis--δ, and (4) a parameter that determines the velocity at large
angles from the axis of symmetry--ηquadrature (See Thomsen, "Weak
Elastic Anisotropy", Geophysics. vol. 51, no. 10, October 1986 and
Alkhalifah and Tsvankin, "Velocity analysis for transversely isotropic
Media", Geophysics, vol. 60, 1550-1566, 1995).

[0005] Some TTI earth models also use anisotropic parameter ε to
describe the propagation of waves in an anisotropic medium. Parameter
ε satisfies the following relationship
η=(ε-δ)/(1+2δ). S (shear) wave velocity is
required to completely describe a TTI earth model, but in P-wave
processing, S-wave velocity is usually obtained using an empirical
relationship with P-wave velocity.

[0006] Usually a TTI earth model is a three directional model. Each point
in the model is described by its coordinates and the values of
anisotropic parameters. In certain situations, only a few quantities of
anisotropic parameters may be needed to fully define a model if the
properties of the anisotropic medium do not change from point to point.
However, in most situations, the TTI earth model requires a large number
of spatially varying values of anisotropic parameters to accurately
define the model.

[0007] The anisotropic parameters of a TTI earth model may be directly
measured from core data. However, drilling a well and coring are very
expensive processes and direct measurements are only possible at very few
well locations. For 3D imaging, it is desirable to determine the
anisotropic parameters of the TTI earth model using also laterally
extended data.

SUMMARY

[0008] In an aspect of the invention, there is provided a computer
implemented method for determining values of anisotropic model parameters
of a Tilted Transversely Isotropic (TTI) Earth model, the anisotropic
parameters including P-wave velocity (Vp0) along a tilted symmetry
axis, the Thomsen anisotropy parameters δ and ε (or
η=(ε-δ)/(1+2δ)) representative of variations of
wave velocities as a function of wave propagation angle from the symmetry
axis, the method including: acquiring input data for a geological volume
of interest; determining a theoretical relationship between the input
data and the anisotropic model parameters; and calculating the values of
the anisotropic model parameters at each of a plurality of subsurface
locations in the geological volume of interest based on the theoretical
relationships and the input data using workflows involving iterative or
sequential combinations of processes including input data preprocessing,
conventional tomographic inversion, three dimensional tomographic
inversion based on a tilted transversely isotropic model, and three
dimensional pre-stack depth migration using a tilted transversely
isotropic model.

[0009] In another aspect of the invention, there is provided a computer
product having machine executable instructions, the instructions being
executable by a machine to perform a tomographic inversion method for
determining values of anisotropic parameters of a TTI earth model, the
anisotropic parameters including P-wave velocity (Vp0) along a
tilted symmetry axis, the Thomsen anisotropy parameters δ and
ε (or η=(ε-δ)/(1+2δ) representative of a
variation of wave velocities as a function of wave propagation angle from
the symmetry axis, the method including determining a relationship
between input data and the anisotropic parameters, the input data being
acquired for a geological volume of interest; and calculating the values
of the anisotropic parameters at each of a plurality of subsurface
locations in the geological volume of interest based on the relationship
and the input data using workflows involving iterative or sequential
combinations of processes including input data preprocessing, three
dimensional tomographic inversion, and three dimensional TTI pre-stack
depth migration.

[0010] A computer implemented method for determining values of anisotropic
parameters of a three dimensional TTI earth model, the anisotropic
parameters including P-wave velocity (Vp0) along a tilted symmetry
axis, the Thomsen anisotropy parameters δ, ε (or
η=(ε-δ)/(1+2δ)) representative of variation of
wave velocities as a function of wave propagation angle from the symmetry
axis, the method including (a) obtaining an initial migration velocity
model that substantially flattens common-imaging-point gathers and
substantially ties seismic data to well data, the initial migration
velocity model including initial values Vp0, δ, ε (or
η) at each of a plurality of subsurface locations in a geological
volume of interest; (b) inputting checkshot data, or VSP data, or both
checkshot and VSP data at well locations into a three dimensional
tomographic inversion to determine updated values of Vp0 near the
well locations, the values of Vp0 being updated by a correction
ΔVp0, wherein Vp0=Vp0+ΔVp0; (c)
determining an incremental improvement Δδ to δ(initial)
using the relative change Δδ=(ΔVp0)/Vp0; (d)
extrapolating the relative change Δδ from near-well locations
to the entire three dimensional TTI earth model at each of the plurality
of subsurface locations, taking into account geological consistency and
regularization, to determine updated values of δ, wherein
δ=δ+Δδ; (e) determining updated values of
Vp0=Vp0 (1-Δδ) using the extrapolated three
dimensional Δδ at each of the plurality of subsurface
locations and obtaining the three dimensional extended incremental update
ΔVp0=-Δδ Vp0; (f) inputting
near-to-mid-offset/angle residual moveout information in the
common-imaging-point gathers obtained with an improved migration velocity
model that is defined with the updated values of Vp0 and δ
into a TTI tomographic inversion process to further provide updated
values of δ at each of the plurality of subsurface locations; and
(g) inputting near-to-far-offset/angle residual moveout information in
the common-imaging-point gathers obtained with the improved model the TTI
tomographic inversion process to provide updated values of η at each
of the plurality of subsurface locations.

[0011] These and other objects, features, and characteristics of the
present invention, as well as the methods of operation and functions of
the related elements of the structure and the combination of parts and
economies of manufacture, will become more apparent upon consideration of
the following description and the appended claims with reference to the
accompanying drawings, all of which form a part of this specification,
wherein like reference numerals designate corresponding parts in the
various figures. It is to be expressly understood, however, that the
drawings are for the purpose of illustration and description only and are
not intended as a definition of the limits of the invention. As used in
the specification and in the claims, the singular form of "a", "an", and
"the" include plural referents unless the context clearly dictates
otherwise.

BRIEF DESCRIPTION OF THE DRAWINGS

[0012] FIG. 1 shows a method for determining values of anisotropic model
parameters of a Tilted Transversely Isotropic (TTI) Earth model in
accordance with an embodiment of the invention;

[0013] FIG. 2 shows a workflow for determining values of anisotropic
parameters in accordance with an embodiment of the invention; and

[0014] FIG. 3 shows various scenarios that can be used in the workflow of
FIG. 2, in accordance with an embodiment of the invention.

DETAILED DESCRIPTION

[0015] Embodiments of the invention provide a method for accurately
determining values of anisotropic parameters of a Titled Transversely
Isotropic (TTI) earth model at each of a plurality of locations in a
geological volume of interest. In an embodiment, the estimated
anisotropic parameters are calibrated with direct measurements at well
locations. Embodiments of the invention also relate to a computer product
having machine executable instructions, the instructions being executable
by a machine to perform a method for determining values of anisotropic
parameters of a TTI earth model.

[0016] In an embodiment, the method for accurately determining values of
anisotropic parameters of a TTI earth model include three main aspects
which are (1) the use of input data to determine the anisotropic
parameters, (2) the relationship between the input data and the
anisotropic parameters, and (3) the process employed to convert the input
data to values of the anisotropic parameters. The process that is used to
convert input data to anisotropic parameters may be referred to as an
inversion process.

[0017] Referring now to FIG. 1, this figure shows a method 10 for
determining values of anisotropic parameters of a TTI earth model in
accordance with an embodiment of the invention. With the method of FIG.
1, the values of the anisotropic parameters (Vp0, δ, η
and/or ε) are determined for each coordinate (x, y, z) in the
geological volume of interest covered by the TTI earth model. The method
starts at procedure 15 where input data are acquired for a geological
volume of interest. Input data may include, for example, surface seismic
data, vertical seismic profile data, VSP (vertical seismic profile) data,
check shot data, well log data, interpretational data, regional trend, a
priori data, or any combination of the foregoing. The interpretational
data includes picks of horizon and shape.

[0018] Surface seismic data, which record reflections from the earth, have
great coverage and are readily available in most areas. The wave
reflected at different angles is sensitive to anisotropic parameters.
Therefore, surface seismic data can be used as input data for the
anisotropic parameter estimation. The measurable data include
travel-times at different offsets and depth error of migrated gathers at
different offsets/angles. VSP and check shot data, which record the
direct arrival of waves from different directions, are also sensitive to
anisotropic parameters. The measurement is travel-time at different
borehole locations. As known in the art, VSP data are acquired by
positioning the wave receiver down the borehole and the wave emitter near
the surface. Check shot data are acquired by positioning both the wave
emitter and receiver down the borehole.

[0019] After acquiring input data, the method 10 proceeds to procedure 20
where a theoretical relationship is determined between the input data and
the anisotropic parameters. The ray theory can be used to derive the
theoretical relationship between the recorded travel-time (input data)
and anisotropic parameters of the TTI earth model. Specifically, the
theoretical relationship between the input data and the anisotropic
parameters can be obtained by first implementing the TTI specification
techniques described in Thomsen, "Weak elastic anisotropy," Geophysics,
vol. 51, pgs. 1954-66 (1986); and/or Alkahalifah, et al., "Velocity
analysis for transversely isotropic media," Geophysics, vol. 60, pgs.
1550-1566 (1995) to specify anisotropic Earth model velocities. The
anisotropic ray tracing is then performed by one or more techniques
described in Cerveny, "Seismic rays and ray intensities in inhomogeneous
anisotropic media," Geophysical Journal, vol. 29, pgs. 1-13 (1972) and/or
Gajewski et al., "Vertical seismic profile synthetics by dynamic ray
tracing in laterally varying layered anisotropic structures," Journal of
Geophysics Res., vol. 95, pgs. 11301-11315, (1990). Additional
information regarding the use of ray theory to derive the theoretical
relationship between the input data and the anisotropic parameters can be
gleaned from U.S. patent application Ser. No. 12/079,170, System and
Method for Migrating Seismic Data, filed on Mar. 24, 2008.

[0020] After determining the theoretical relationship between the input
data and the anisotropic parameters, the method proceeds to procedure 25
where the values of the anisotropic model parameters at each of a
plurality of subsurface locations in the geological volume of interest
are calculated based on the theoretical relationships and the input data.
In an embodiment, the calculation of the anisotropic parameters uses
workflows involving iterative or sequential combinations of processes
including input data preprocessing, conventional tomographic inversion or
three dimensional tomographic inversion based on a tilted transversely
isotropic model, and three dimensional pre-stack depth migration using a
tilted transversely isotropic model. Further, the iterative or sequential
combinations of processes may include interpretive picking.

[0021] The estimation of the anisotropic parameters of procedure 25 is a
computational process which uses the recorded data and a theoretical
relationship to solve the model parameters. This process is called
tomographic inversion. The inversion can be performed differently using
various workflows. For example, the anisotropic parameters may be
estimated sequentially (i.e. one at a time) or, preferably, various
parameters may be estimated simultaneously. It is also possible to use
one type of data to estimate one or multiple parameters. In another
embodiment, all available input data are used to estimate all anisotropic
parameters simultaneously. Further, the values of the anisotropic model
parameters can be determined using various combinations of inversions
with full sets or subsets of the input data and full sets or subsets of
the output data.

[0022] The tomographic inversion of procedure 25 is recursive and
iterative, and may use three dimensional tilted transversely isotropic
ray tracing to model three dimensional wave propagation. Input data
preprocessing may include normalizing the input before performing the
three dimensional TTI tomographic inversion. Input data preprocessing may
also include, prior to performing the three dimensional TTI tomographic
inversion, migrating the seismic data using 3D TTI prestack depth
migration algorithms, sorting the migrated seismic data according to
their subsurface location and their migration offset/angle into
common-image-point (CIP) gathers, and selecting and quantifying residual
moveouts in the common-image-point gathers domain. TTI prestack depth
migration algorithms are known in the art.

[0023] The values of the anisotropic parameters are calculated by
iteratively selecting values of Vp0, δ and η.
Specifically, the values of the anisotropic parameters are calculated
until (a) seismic image positions of subsurface structures in the
geological volume of interest tie their spatial positions recognized in
drilled wells and (b) residual moveouts in common-imaging-point gathers
are minimum at every point in the model. The subsurface structures
include rock boundaries.

[0024] Referring now to FIG. 2, this figure shows a workflow 100 for
determining values of anisotropic parameters of a Tilted Transversely
Isotropic (TTI) earth model in accordance with an embodiment of the
invention. The iterative workflow can begin at procedure 105 where three
dimensional seismic reflection data are measured. The workflow then
proceeds to procedure 110 where the seismic reflection data are used to
determine an initial TTI earth model for the geological volume of
interest. The initial TTI earth model may be determined using various
means. For example, the initial TTI earth model can be determined from
velocity information and various processing performed on measured data as
well as knowledge, either regionally or globally, about the area in which
the geological volume of interest is located. Furthermore, the initial
TTI model building also includes estimating subsurface reflector
structural dips and obtaining TTI symmetry axes in the geological volume
of interest. In the initial TTI earth model, a value of each parameter
Vp0, δ and η is provided at each of a plurality of
locations (x, y, z) in the geological volume of interest.

[0025] After determining an initial TTI earth model, the workflow proceeds
to block 115 and procedure 120 where well data (block 115) are used to
perform a three dimensional (3D) TTI tomography (procedure 120) based on
ray tracing to update the value of Vp0 of the TTI earth model near
the well(s). Well data may include VSP data or checkshot data, which
represent information about wave velocities in the well(s). The updated
values of Vp0 in the geological volume of interest near the well(s)
corresponds to Vp0(initial)+ΔVp0. With this operation,
the values of Vp0 near the well(s) are updated in the initial TTI
earth model. Specifically, the VSP data and checkshot data are used in
the embodiment of FIG. 2 to tie the seismic data to the well data.

[0026] After updating the values of Vp0 near the well(s), the
workflow 100 proceeds to procedure 125 where the initial values of
δ near the well(s) are updated using the updated values of Vp0
according to the following transformation
Δδ=ΔVp0/Vp0 with
δ(updated)=δ(initial)+Δδ. The transformations of
procedures 110-125 enable one to refine the initial 3D TTI earth model by
providing a more accurate TTI model near the well(s).

[0027] Once the initial TTI earth model has been updated near the well(s)
with the updated values of Vp0 and δ, the workflow 100
proceeds to procedure 130 where the remaining points in the model, i.e.
the points outside the location(s) near the well(s), are populated. In
the embodiment of FIG. 2, the remaining points in the model are populated
using a three dimensional extrapolation process and the updated values of
Vp0 and δ near the well(s). The three dimensional
extrapolation process is performed by first extrapolating δ or
Δδ in the remaining locations of the model and then obtaining
a three dimensional update of Vp0 using the transformation
ΔVp0=-Δδ*Vp0. It will be appreciated that
various types of algorithms can be used to extrapolate the values of
δ or Δδ outside the well locations and to obtain
extended updates of Vp0 and δ. The result of the extrapolation
process (see block 135) provides a three dimensional TTI earth model with
well tie having updated values of Vp0 and δ at each point of
the model. The three dimensional TTI earth model of block 135 includes
the following parameters at each point (x, y, z) of the model:
Vp0(initial)+ΔVp0, δ(initial)+Δδ and
η(initial).

[0028] The model determined at block 135 is used to perform a three
dimensional (3D) pre-stack TTI depth imaging of seismic reflection data
at procedure 140. With this procedure, seismic reflection data that have
been recorded for the geological volume of interest are
processed/migrated by the model to provide a new image of the sub-surface
of the geological volume of interest. As will be appreciated, the
migration of seismic data will typically enable an image to be formed of
the geological volume of interest from the migrated seismic data that is
a more accurate depiction of the geological features present in the
seismic volume of interest than an image formed from migrated data with
an isotropic algorithm. The three dimensional pre-stack TTI depth imaging
of seismic reflection data of procedure 140 provides common image point
gathers (block 145). As known in the art, common image point gathers
correspond to the migrated seismic data that end up at a same image
position in the geological volume of interest.

[0029] Once the common image point gathers have been identified by the 3D
pre-stack TTI depth imaging analysis, the workflow 100 then proceeds to
procedures 150 and/or 155 wherein a residual moveout analysis is
performed. As known in the art, residual moveout analysis is a basic step
in velocity model refinement. The analysis is generally carried out using
semblances for a grid of inline and xline locations. One semblance panel
is computed for each common-image-point gather as a function of depth and
offset. The residual moveout analysis on the common image point gathers
is carried out for the near to mid offsets at procedure 150 while the
residual moveout analysis on the common image point gathers is carried
out for the near to far mid offsets at procedure 155. Near and far
offsets correspond, respectively, to a small and a large distance between
the source and the receiver.

[0030] It will be appreciated that the procedures 150, 155 of FIG. 2 are
intended to cover multiple moveout analyses. That is, the residual
moveout analysis of FIG. 2 can be performed only for the near to mid
offsets (procedure 150) or only for the near to far offsets (procedure
155) in embodiments of the invention. Alternatively, in an embodiment of
the invention, and as shown in FIG. 2, the residual moveout analysis can
be performed for both the near to mid offsets and the near to far offsets
simultaneously.

[0031] The residual moveout analysis provides updated wave velocities
Vp0 for each point of the 3D TTI model defined at procedure 135.
After determining the updated velocities Vp0 for each point of the
model, a determination is made at blocks 165a and/or 165b as to whether
convergence is obtained, i.e. whether the results of the residual moveout
analyses of procedures 150 and/or procedure 155 is/are within a
predetermined tolerance. If convergence is obtained, the workflow 100
ends at block 166.

[0032] If convergence is not obtained, the workflow 100 proceeds to block
170 and/or block 180 wherein it is determined whether the values of
δ and/or η should be updated for each point of the model. If
the result of the inquiry is positive at block 170, the workflow proceeds
to procedure 185 where a three dimensional (3D) TTI tomography is
performed to update the value of δ for each point of the model
using the results of the residual moveout analysis for the near to mid
offsets. It is greatly beneficial to update the values of δ only
because such an update does not modify the values of wave velocities
Vp0 in the model. As a result, the tie between well and seismic data
that was previously obtained with the model constructed at block 135 is
not qualitatively destroyed in a low dip setting. In that way, it is
possible to obtain a faster and more robust conversion for the values of
the TTI model.

[0033] Alternatively or additionally, it is determined whether the values
of η should be updated for each point of the model. If the result of
the inquiry is positive at block 180, the workflow proceeds to procedure
195 where a three dimensional (3D) TTI tomography is performed to update
the value of η for each point of the model using the results of the
residual moveout analysis for the far to mid offsets.

[0034] If the values of δ and/or η are not updated, it is
determined whether the values of wave velocities Vp0 should be
updated (block 175). If the result of the inquiry is negative, the
workflow ends at block 166. If the result of the inquiry is positive, the
workflow proceeds to procedure 190 where a 3D TTI tomography is performed
to update the values of velocities Vp0 in the geological volume of
interest.

[0035] The results of the 3D TTI tomography analyses of procedures 185,
190 and 195 provide a new three dimensional (3D) TTI model at block 197
with updated values of δ, Vp0 and η at each point (x, y,
z) of the model. It is then determined whether convergence is obtained
for the updated values of δ, Vp0 and η in the new model of
block 197 (block 198). Various tests may be used to determine whether
convergence is obtained. For example, it is determined whether the
obtained values are below a predetermined threshold. If the result of the
inquiry is positive, the workflow of FIG. 2 ends at procedure 199. If the
result of the inquiry is negative, the workflow 100 proceeds back to
block 115 where the well data are used to perform a three dimensional
(3D) TTI tomography (procedure 120) based on ray tracing to update the
value of Vp0 of the TTI earth model obtained at block 197 near the
well(s). Thus, the TTI earth model obtained at block 197 is used as a new
initial model for the subsequent iteration. In an embodiment, this new
initial TTI earth model is more refined than the initial model used at
the first iteration in that it substantially flattens
common-imaging-point gathers and substantially ties seismic data to well
data. After determining a modified model with updated values of Vp0
near the well(s), the workflow 100 proceeds to procedures 125-198 where
updated values of δ and/or Vp0 and/or η are determined.
The workflow is then iterated until convergence is obtained for the
values of δ and/or Vp0 and/or η (i.e. the values of
δ and/or Vp0 and/or η substantially do not change between
two subsequent iterations).

[0036] It will be appreciated that the workflow of FIG. 2 is intended to
encompass several scenarios for optimizing the three dimensional (3D) TTI
model. Referring now to FIG. 3, this figure shows the various scenarios
that can be applied at each iteration. In the first scenario, δ is
the only parameter that is updated at each location in the model. The
first scenario is preferred because the update of δ does not change
the values of Vp0 in a low dip setting and, as a result, the
quantitative tie between the well data and the seismic data is not
destroyed during the optimization of δ. In the second scenario,
parameters δ and Vp0 are updated. In the third scenario,
parameters δ and η are updated. In the fourth scenario, the
velocities Vp0 are updated. In the fifth scenario, Vp0 and
η are updated. In the sixth scenario, η is updated and, in the
seventh scenario, δ, η and Vp0 are updated.

[0037] It will be appreciated that the one or more parameters updated at a
given iteration may not be the same as the one or more parameters updated
at a subsequent iteration. In other words, the anisotropic parameters
(δ, η and Vp0) optimized at each iteration may be
different. Thus, in an embodiment, it is envisioned that various
scenarios could be used to optimize the three dimensional TTI model.
Further, it is envisioned that after optimizing a first parameter, e.g.
δ according to the first scenario, the workflow of FIG. 2 may be
pursued to optimize a second parameter, e.g. η, and then a third
parameter, e.g. Vp0. Therefore, the parameters δ, η, and
Vp0 can be optimized simultaneously or sequentially. Further, it
will be appreciated that the selection of the parameters updated at each
iteration is highly dependent on the maturity of the overall model
building process that includes both depth imaging and tomography and the
type and the quality of well and seismic data that are used to construct
the 3D TTI model.

[0038] Although the invention has been described in detail for the purpose
of illustration based on what is currently considered to be the most
practical and preferred embodiments, it is to be understood that such
detail is solely for that purpose and that the invention is not limited
to the disclosed embodiments, but, on the contrary, is intended to cover
modifications and equivalent arrangements that are within the spirit and
scope of the appended claims.

[0039] It will be appreciated that the different acts involved in
determining values of anisotropic model parameters of a Tilted
Transversely Isotropic (TTI) Earth model may be executed according to
machine executable instructions or codes. These machine executable
instructions may be embedded in a data storage medium. A processor may be
configured to execute the instructions.

[0040] Software functionalities of a computer system involving
programming, including executable codes, may be used to implement the
above described model. The software code may be executable by a
general-purpose computer. In operation, the code and possibly the
associated data records may be stored within a general-purpose computer
platform. At other times, however, the software may be stored at other
locations and/or transported for loading into an appropriate
general-purpose computer system. Hence, the embodiments discussed above
involve one or more software or computer products in the form of one or
more modules of code carried by at least one machine-readable medium.
Execution of such codes by a processor of the computer system enables the
platform to implement the functions in essentially the manner performed
in the embodiments discussed and illustrated herein.

[0041] As used herein, terms such as computer or machine "readable medium"
refer to any medium that participates in providing instructions to a
processor for execution. A computer or a machine "readable medium" may be
broadly termed a "computer product." Such a medium may take many forms,
including but not limited to, non-volatile media, volatile media, and
transmission media. Non-volatile media include, for example, optical or
magnetic disks, such as any of the storage devices in any computer(s)
operating as discussed above. Volatile media include dynamic memory, such
as the main memory of a computer system. Physical transmission media
include coaxial cables, copper wires and fiber optics, including the
wires that comprise a bus within a computer system. Carrier-wave
transmission media can take the form of electric or electromagnetic
signals, or acoustic or light waves such as those generated during radio
frequency (RF) and infrared (IR) data communications. Common forms of
computer-readable media therefore include, for example: a floppy disk, a
flexible disk, hard disk, magnetic tape, any other magnetic medium, a
CD-ROM, DVD, any other optical medium, less commonly used media such as
punch cards, paper tape, any other physical medium with patterns of
holes, a RAM, a PROM, and EPROM, a FLASH-EPROM, any other memory chip or
cartridge, a carrier wave transporting data or instructions, cables or
links transporting such a carrier wave, or any other medium from which a
computer can read or send programming codes and/or data. Many of these
forms of computer readable media may be involved in carrying one or more
sequences of one or more instructions to a processor for execution.

[0042] It is to be understood that the present invention contemplates
that, to the extent possible, one or more features of any embodiment can
be combined with one or more features of any other embodiment.